Mixing
Unitaries in Banach *-algebras
Clash Royale CLAN TAG#URR8PPP
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We know that the norm of a unitary in a unital $C^*$-algebra is one. Also, in a unital Banach algebra A, $u in A$ is defined to be a unitary if $|u| = |u^-1| =1$. I tried to prove it for the unitaries in unital Banach $*$-algebra, but not able to get an answer. Is it true that a unitary in a unital Banach$*$-algebra has norm one? Is there some counter example?
oa.operator-algebras
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Ranjana Jain
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up vote
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We know that the norm of a unitary in a unital $C^*$-algebra is one. Also, in a unital Banach algebra A, $u in A$ is defined to be a unitary if $|u| = |u^-1| =1$. I tried to prove it for the unitaries in unital Banach $*$-algebra, but not able to get an answer. Is it true that a unitary in a unital Banach$*$-algebra has norm one? Is there some counter example?
oa.operator-algebras
asked 4 hours ago
Ranjana Jain
62
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Ranjana Jain is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
We know that the norm of a unitary in a unital $C^*$-algebra is one. Also, in a unital Banach algebra A, $u in A$ is defined to be a unitary if $|u| = |u^-1| =1$. I tried to prove it for the unitaries in unital Banach $*$-algebra, but not able to get an answer. Is it true that a unitary in a unital Banach$*$-algebra has norm one? Is there some counter example?
oa.operator-algebras
asked 4 hours ago
Ranjana Jain
62
New contributor
Ranjana Jain is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
We know that the norm of a unitary in a unital $C^*$-algebra is one. Also, in a unital Banach algebra A, $u in A$ is defined to be a unitary if $|u| = |u^-1| =1$. I tried to prove it for the unitaries in unital Banach $*$-algebra, but not able to get an answer. Is it true that a unitary in a unital Banach$*$-algebra has norm one? Is there some counter example?
oa.operator-algebras
oa.operator-algebras
asked 4 hours ago
Ranjana Jain
62
New contributor
Ranjana Jain is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Ranjana Jain
62
New contributor
Ranjana Jain is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 4 hours ago
asked 4 hours ago
Ranjana Jain
62
New contributor
Ranjana Jain is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Ranjana Jain
62
asked 4 hours ago
Ranjana Jain
62
62
New contributor
Ranjana Jain is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Ranjana Jain is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Ranjana Jain is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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2 Answers
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The answer is no even for a particular unitisation of the algebra of compact operators. We consider the Banach algebra $widetildeK:= lambda Id + k: ktext is compact$, given with natural involution and norm $|lambda Id + k| := |lambda| + |k|_op$ (it's not equal to the operator norm!); it is a unital Banach $ast$-algebra. The element $left[beginarraycccc 0 & 1 & 0 & cdots \ 1 & 0 & 0 & cdots \ 0 & 0 & 1 & cdots \ vdots & vdots & vdots &ddots endarrayright]$ is unitary, but it's norm is equal to $3$.
answered 52 mins ago
Mateusz Wasilewski
2,530816
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up vote
1
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If your question is:
suppose that $u^* u=1$. Must $u$ have norm one?
then no, as already the unit may be counterexample. Indeed, there exist non-unital semigroups $S$ whose semigroup algebras $ell_1(S)$ are unital but the the unit has norm greater than one.
answered 3 hours ago
Tomek Kania
5,97322152
Thanks for the reply, but here we assume (in unital Banach $*$-algebras) that unit has norm 1
– Ranjana Jain
3 hours ago
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
The answer is no even for a particular unitisation of the algebra of compact operators. We consider the Banach algebra $widetildeK:= lambda Id + k: ktext is compact$, given with natural involution and norm $|lambda Id + k| := |lambda| + |k|_op$ (it's not equal to the operator norm!); it is a unital Banach $ast$-algebra. The element $left[beginarraycccc 0 & 1 & 0 & cdots \ 1 & 0 & 0 & cdots \ 0 & 0 & 1 & cdots \ vdots & vdots & vdots &ddots endarrayright]$ is unitary, but it's norm is equal to $3$.
answered 52 mins ago
Mateusz Wasilewski
2,530816
add a comment |Â
up vote
3
down vote
The answer is no even for a particular unitisation of the algebra of compact operators. We consider the Banach algebra $widetildeK:= lambda Id + k: ktext is compact$, given with natural involution and norm $|lambda Id + k| := |lambda| + |k|_op$ (it's not equal to the operator norm!); it is a unital Banach $ast$-algebra. The element $left[beginarraycccc 0 & 1 & 0 & cdots \ 1 & 0 & 0 & cdots \ 0 & 0 & 1 & cdots \ vdots & vdots & vdots &ddots endarrayright]$ is unitary, but it's norm is equal to $3$.
answered 52 mins ago
Mateusz Wasilewski
2,530816
add a comment |Â
up vote
3
down vote
up vote
3
down vote
The answer is no even for a particular unitisation of the algebra of compact operators. We consider the Banach algebra $widetildeK:= lambda Id + k: ktext is compact$, given with natural involution and norm $|lambda Id + k| := |lambda| + |k|_op$ (it's not equal to the operator norm!); it is a unital Banach $ast$-algebra. The element $left[beginarraycccc 0 & 1 & 0 & cdots \ 1 & 0 & 0 & cdots \ 0 & 0 & 1 & cdots \ vdots & vdots & vdots &ddots endarrayright]$ is unitary, but it's norm is equal to $3$.
answered 52 mins ago
Mateusz Wasilewski
2,530816
The answer is no even for a particular unitisation of the algebra of compact operators. We consider the Banach algebra $widetildeK:= lambda Id + k: ktext is compact$, given with natural involution and norm $|lambda Id + k| := |lambda| + |k|_op$ (it's not equal to the operator norm!); it is a unital Banach $ast$-algebra. The element $left[beginarraycccc 0 & 1 & 0 & cdots \ 1 & 0 & 0 & cdots \ 0 & 0 & 1 & cdots \ vdots & vdots & vdots &ddots endarrayright]$ is unitary, but it's norm is equal to $3$.
answered 52 mins ago
Mateusz Wasilewski
2,530816
answered 52 mins ago
Mateusz Wasilewski
2,530816
answered 52 mins ago
Mateusz Wasilewski
2,530816
answered 52 mins ago
Mateusz Wasilewski
2,530816
2,530816
add a comment |Â
add a comment |Â
up vote
1
down vote
If your question is:
suppose that $u^* u=1$. Must $u$ have norm one?
then no, as already the unit may be counterexample. Indeed, there exist non-unital semigroups $S$ whose semigroup algebras $ell_1(S)$ are unital but the the unit has norm greater than one.
answered 3 hours ago
Tomek Kania
5,97322152
Thanks for the reply, but here we assume (in unital Banach $*$-algebras) that unit has norm 1
– Ranjana Jain
3 hours ago
add a comment |Â
up vote
1
down vote
If your question is:
suppose that $u^* u=1$. Must $u$ have norm one?
then no, as already the unit may be counterexample. Indeed, there exist non-unital semigroups $S$ whose semigroup algebras $ell_1(S)$ are unital but the the unit has norm greater than one.
answered 3 hours ago
Tomek Kania
5,97322152
Thanks for the reply, but here we assume (in unital Banach $*$-algebras) that unit has norm 1
– Ranjana Jain
3 hours ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
If your question is:
suppose that $u^* u=1$. Must $u$ have norm one?
then no, as already the unit may be counterexample. Indeed, there exist non-unital semigroups $S$ whose semigroup algebras $ell_1(S)$ are unital but the the unit has norm greater than one.
answered 3 hours ago
Tomek Kania
5,97322152
If your question is:
suppose that $u^* u=1$. Must $u$ have norm one?
then no, as already the unit may be counterexample. Indeed, there exist non-unital semigroups $S$ whose semigroup algebras $ell_1(S)$ are unital but the the unit has norm greater than one.
answered 3 hours ago
Tomek Kania
5,97322152
answered 3 hours ago
Tomek Kania
5,97322152
answered 3 hours ago
Tomek Kania
5,97322152
answered 3 hours ago
Tomek Kania
5,97322152
5,97322152
Thanks for the reply, but here we assume (in unital Banach $*$-algebras) that unit has norm 1
– Ranjana Jain
3 hours ago
add a comment |Â
Thanks for the reply, but here we assume (in unital Banach $*$-algebras) that unit has norm 1
– Ranjana Jain
3 hours ago
Thanks for the reply, but here we assume (in unital Banach $*$-algebras) that unit has norm 1
– Ranjana Jain
3 hours ago
Thanks for the reply, but here we assume (in unital Banach $*$-algebras) that unit has norm 1
– Ranjana Jain
3 hours ago
add a comment |Â
Ranjana Jain is a new contributor. Be nice, and check out our Code of Conduct.
Ranjana Jain is a new contributor. Be nice, and check out our Code of Conduct.
Ranjana Jain is a new contributor. Be nice, and check out our Code of Conduct.
Ranjana Jain is a new contributor. Be nice, and check out our Code of Conduct.
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