strictly positive element vs positive definite matrix
If $A=prod_{n=1}^{infty}M_{k(n)}(mathbb{C})$,$x=(x_1,cdots,x_n,cdots)$ is strictly positive in $A$,does this mean that each $x_nin M_{k(n)}mathbb{C}$ is a positive definite matrix?
operator-theory operator-algebras c-star-algebras
edited Dec 5 '18 at 17:32
Ethan Bolker
42.1k548111
asked Dec 5 '18 at 17:13
mathrookiemathrookie
826512
add a comment |
If $A=prod_{n=1}^{infty}M_{k(n)}(mathbb{C})$,$x=(x_1,cdots,x_n,cdots)$ is strictly positive in $A$,does this mean that each $x_nin M_{k(n)}mathbb{C}$ is a positive definite matrix?
operator-theory operator-algebras c-star-algebras
edited Dec 5 '18 at 17:32
Ethan Bolker
42.1k548111
asked Dec 5 '18 at 17:13
mathrookiemathrookie
826512
add a comment |
If $A=prod_{n=1}^{infty}M_{k(n)}(mathbb{C})$,$x=(x_1,cdots,x_n,cdots)$ is strictly positive in $A$,does this mean that each $x_nin M_{k(n)}mathbb{C}$ is a positive definite matrix?
operator-theory operator-algebras c-star-algebras
edited Dec 5 '18 at 17:32
Ethan Bolker
42.1k548111
asked Dec 5 '18 at 17:13
mathrookiemathrookie
826512
If $A=prod_{n=1}^{infty}M_{k(n)}(mathbb{C})$,$x=(x_1,cdots,x_n,cdots)$ is strictly positive in $A$,does this mean that each $x_nin M_{k(n)}mathbb{C}$ is a positive definite matrix?
operator-theory operator-algebras c-star-algebras
operator-theory operator-algebras c-star-algebras
edited Dec 5 '18 at 17:32
Ethan Bolker
42.1k548111
asked Dec 5 '18 at 17:13
mathrookiemathrookie
826512
edited Dec 5 '18 at 17:32
Ethan Bolker
42.1k548111
asked Dec 5 '18 at 17:13
mathrookiemathrookie
826512
edited Dec 5 '18 at 17:32
Ethan Bolker
42.1k548111
edited Dec 5 '18 at 17:32
Ethan Bolker
42.1k548111
edited Dec 5 '18 at 17:32
Ethan Bolker
42.1k548111
42.1k548111
asked Dec 5 '18 at 17:13
mathrookiemathrookie
826512
asked Dec 5 '18 at 17:13
mathrookiemathrookie
826512
asked Dec 5 '18 at 17:13
mathrookiemathrookie
826512
826512
add a comment |
add a comment |
1 Answer
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In a unital C$^*$-algebar, "strictly positive" is the same as "positive and invertible", which is exactly "positive definite". So yes, if $x$ is strictly positive then each $x_n$ is positive and invertible, so positive definite.
answered Dec 5 '18 at 17:31
Martin ArgeramiMartin Argerami
124k1176175
add a comment |
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1 Answer
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In a unital C$^*$-algebar, "strictly positive" is the same as "positive and invertible", which is exactly "positive definite". So yes, if $x$ is strictly positive then each $x_n$ is positive and invertible, so positive definite.
answered Dec 5 '18 at 17:31
Martin ArgeramiMartin Argerami
124k1176175
add a comment |
In a unital C$^*$-algebar, "strictly positive" is the same as "positive and invertible", which is exactly "positive definite". So yes, if $x$ is strictly positive then each $x_n$ is positive and invertible, so positive definite.
answered Dec 5 '18 at 17:31
Martin ArgeramiMartin Argerami
124k1176175
add a comment |
In a unital C$^*$-algebar, "strictly positive" is the same as "positive and invertible", which is exactly "positive definite". So yes, if $x$ is strictly positive then each $x_n$ is positive and invertible, so positive definite.
answered Dec 5 '18 at 17:31
Martin ArgeramiMartin Argerami
124k1176175
In a unital C$^*$-algebar, "strictly positive" is the same as "positive and invertible", which is exactly "positive definite". So yes, if $x$ is strictly positive then each $x_n$ is positive and invertible, so positive definite.
answered Dec 5 '18 at 17:31
Martin ArgeramiMartin Argerami
124k1176175
answered Dec 5 '18 at 17:31
Martin ArgeramiMartin Argerami
124k1176175
answered Dec 5 '18 at 17:31
Martin ArgeramiMartin Argerami
124k1176175
answered Dec 5 '18 at 17:31
Martin ArgeramiMartin Argerami
124k1176175
124k1176175
add a comment |
add a comment |
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