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The inner structure of finite-dimensional $C^*$-algebras

Two isomorphic finite-dimensional $C^*$-algebras with infinite eigenspacesSpatial tensor product of algebras and normed spacesConnection between Stinespring's factorization theorem and the spectral theorem for bounded operatorsFor a finite dimensional Hilbert space, is every automorphism “approximately inner”?$M_2(mathcalK(mathbbH)) cong mathcalK(mathbbH oplus mathbbH )$?$B$ stably finite simple $C^*$-algebra $Rightarrow$ $Botimes mathcalK$ contains no infinite projectionsminimal projections in finite dimensional von Neumann algebrastracial state on a unital infinite dimensional simple $C^*$ algebraTwo isomorphic $C^*$-algebras. What is the isomorphism between corresponding Hilbert spaces?Two isomorphic finite-dimensional $C^*$-algebras with infinite eigenspacesReference request unital normal *-homomorphisms $B(H) to B(K)$

1

$begingroup$

This is just to complete the questions about the inner structure of finite-dimensional $C^*$-algebras. Please correct the statements and help me with the appropriate links if it is not very difficult. Maybe I am wrong somewhere.

Let $H$ be a Hilbert space with a finite or countable basis. Let $mathscrA$ be a finite-dimensional unital $C^*$-algebra of operators acting on $H$. One of the diamonds, the Wedderburn theorem states that $mathscrA$ is a direct sum of simple matrix algebras
$$
mathscrAcongbigoplus_n=1^NmathbbC^m_ntimes m_n.
$$
But we need a bit more, namely the characterization of its inner structure in terms of unitary transformations of Hilbert spaces. Let $mathcalE^n_jj$ be the orthogonal projectors corresponding to the diagonal matrix units in $mathbbC^m_ntimes m_n$. I think that the following statements are true
$$
i) rm rank(mathcalE^n_11)=...=rm rank(mathcalE^n_m_nm_n), n=1,...,N;
$$
$$
ii) sum_n=1^Nm_nrm rank(mathcalE^n_11)=dim H.
$$
The ranks can be finite or infinite. Now, I think that the set of pairs of numbers
$$
rm struc(mathscrA)=(m_n,rm rank(mathcalE^n_11))_n=1^N
$$
is the complete characteristic of the inner structure of $mathscrA$. Namely, a unital $C^*$-alebra $mathscrA_1$ of operators acting on a Hilbert space $H_1$ has the same $rm struc(mathscrA)=rm struc(mathscrA_1)$ if and only if there is a unitary $mathcalU:Hto H_1$ such that $mathscrA_1=mathcalUmathscrAmathcalU^-1$. Moreover, for any finite set of pairs $ssubsetmathbbNtimesoverlinemathbbN$ there is a finite-dimensional unital $C^*$-algebra $mathscrA$ of operators acting on some Hilbert space such that $rm struc(mathscrA)=s$.

I think the idea of the Proof of this statement is the same as in the comment Two isomorphic finite-dimensional $C^*$-algebras with infinite eigenspaces or can be extracted from the Proof of Wedderburn theorem directly. I am sorry for the multiple more or less basic questions about the structure of $C^*$-algebras, but I know the people for whom this information can be useful.

c-star-algebras

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edited yesterday

AAK

asked yesterday

AAKAAK

566

$endgroup$

add a comment | 

1

$begingroup$

This is just to complete the questions about the inner structure of finite-dimensional $C^*$-algebras. Please correct the statements and help me with the appropriate links if it is not very difficult. Maybe I am wrong somewhere.

Let $H$ be a Hilbert space with a finite or countable basis. Let $mathscrA$ be a finite-dimensional unital $C^*$-algebra of operators acting on $H$. One of the diamonds, the Wedderburn theorem states that $mathscrA$ is a direct sum of simple matrix algebras
$$
mathscrAcongbigoplus_n=1^NmathbbC^m_ntimes m_n.
$$
But we need a bit more, namely the characterization of its inner structure in terms of unitary transformations of Hilbert spaces. Let $mathcalE^n_jj$ be the orthogonal projectors corresponding to the diagonal matrix units in $mathbbC^m_ntimes m_n$. I think that the following statements are true
$$
i) rm rank(mathcalE^n_11)=...=rm rank(mathcalE^n_m_nm_n), n=1,...,N;
$$
$$
ii) sum_n=1^Nm_nrm rank(mathcalE^n_11)=dim H.
$$
The ranks can be finite or infinite. Now, I think that the set of pairs of numbers
$$
rm struc(mathscrA)=(m_n,rm rank(mathcalE^n_11))_n=1^N
$$
is the complete characteristic of the inner structure of $mathscrA$. Namely, a unital $C^*$-alebra $mathscrA_1$ of operators acting on a Hilbert space $H_1$ has the same $rm struc(mathscrA)=rm struc(mathscrA_1)$ if and only if there is a unitary $mathcalU:Hto H_1$ such that $mathscrA_1=mathcalUmathscrAmathcalU^-1$. Moreover, for any finite set of pairs $ssubsetmathbbNtimesoverlinemathbbN$ there is a finite-dimensional unital $C^*$-algebra $mathscrA$ of operators acting on some Hilbert space such that $rm struc(mathscrA)=s$.

I think the idea of the Proof of this statement is the same as in the comment Two isomorphic finite-dimensional $C^*$-algebras with infinite eigenspaces or can be extracted from the Proof of Wedderburn theorem directly. I am sorry for the multiple more or less basic questions about the structure of $C^*$-algebras, but I know the people for whom this information can be useful.

c-star-algebras

share|cite|improve this question

edited yesterday

AAK

asked yesterday

AAKAAK

566

$endgroup$

add a comment | 

$begingroup$

This is just to complete the questions about the inner structure of finite-dimensional $C^*$-algebras. Please correct the statements and help me with the appropriate links if it is not very difficult. Maybe I am wrong somewhere.

Let $H$ be a Hilbert space with a finite or countable basis. Let $mathscrA$ be a finite-dimensional unital $C^*$-algebra of operators acting on $H$. One of the diamonds, the Wedderburn theorem states that $mathscrA$ is a direct sum of simple matrix algebras
$$
mathscrAcongbigoplus_n=1^NmathbbC^m_ntimes m_n.
$$
But we need a bit more, namely the characterization of its inner structure in terms of unitary transformations of Hilbert spaces. Let $mathcalE^n_jj$ be the orthogonal projectors corresponding to the diagonal matrix units in $mathbbC^m_ntimes m_n$. I think that the following statements are true
$$
i) rm rank(mathcalE^n_11)=...=rm rank(mathcalE^n_m_nm_n), n=1,...,N;
$$
$$
ii) sum_n=1^Nm_nrm rank(mathcalE^n_11)=dim H.
$$
The ranks can be finite or infinite. Now, I think that the set of pairs of numbers
$$
rm struc(mathscrA)=(m_n,rm rank(mathcalE^n_11))_n=1^N
$$
is the complete characteristic of the inner structure of $mathscrA$. Namely, a unital $C^*$-alebra $mathscrA_1$ of operators acting on a Hilbert space $H_1$ has the same $rm struc(mathscrA)=rm struc(mathscrA_1)$ if and only if there is a unitary $mathcalU:Hto H_1$ such that $mathscrA_1=mathcalUmathscrAmathcalU^-1$. Moreover, for any finite set of pairs $ssubsetmathbbNtimesoverlinemathbbN$ there is a finite-dimensional unital $C^*$-algebra $mathscrA$ of operators acting on some Hilbert space such that $rm struc(mathscrA)=s$.

I think the idea of the Proof of this statement is the same as in the comment Two isomorphic finite-dimensional $C^*$-algebras with infinite eigenspaces or can be extracted from the Proof of Wedderburn theorem directly. I am sorry for the multiple more or less basic questions about the structure of $C^*$-algebras, but I know the people for whom this information can be useful.

c-star-algebras

share|cite|improve this question

edited yesterday

AAK

asked yesterday

AAKAAK

566

$endgroup$

share|cite|improve this question

edited yesterday

AAK

asked yesterday

AAKAAK

566

share|cite|improve this question

edited yesterday

AAK

asked yesterday

AAKAAK

566

asked yesterday

AAKAAK

566

asked yesterday

AAKAAK

566